DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2011.16.2iDYNAMICAL SYSTEMS SERIES BVolume 16, Number 2, September 2011 pp. i–iv

PREFACE

This issue of Discrete and Continuous Dynamical Systems–Series B, is dedicatedto our professor and friend, Qishao Lu, on the occasion of his 70th birthday andin honor of his important and fundamental contributions to the fields of appliedmathematics, theoretical mechanics and computational neurodynamics. His pleas-ant personality and ready helpfulness have won our hearts as his admirers, students,and friends.

Qishao Lu was born in Shaoguan City, Guangdong Province, China on June 13,1940. He graduated from the Beijing University of Aeronautics and Astronautics(BUAA) in aerodynamics in 1960, since then he had worked in his mother universityas lecturer, associate professor and full professor. In the first three years of 1980s, hevisited the Department of Mathematics of Purdue University as a visiting professor.After his return in 1983, Qishao founded the Center for Nonlinear Dynamics and

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organized related seminars in BUAA. Some famous mathematicians were invited todeliver lectures there in late 1980s, including L. Cesari, J. E. Marsden, M. Golubit-sky, W. F. Langford and Shui-Nee Chow etc., and a close academic bridge was builtbetween BUAA and the outside world. From then on he has established extensivecollaboration in research work with more well-known mathematicians, physicistsand scholars in China and other countries. He holds a number of visiting positionsin universities and institutes of United States, United Kingdom, Russia, Canada,Germany and other countries or regions. He was the Chair of the Department ofApplied Mathematics and the Director of the Center for Nonlinear Dynamics ofBUAA, and served as the chair (or co-chair) of organizing committees for morethan ten international conferences.

Qishao Lu has been one of the most productive and distinguished scientists inBUAA over a period of fifty years. His research work covers a wide range of subjectson nonlinear ordinary and partial differential equations, dynamical systems, as wellas applications in theoretical physics, mechanics, biology and engineering. He hasprovided a number of effective mathematical techniques for the study of stability,bifurcation and chaos in higher dimensional nonlinear systems, time-delay systemsand non-smooth systems, which are also successfully applied to mathematical mod-eling, theoretical analysis and numerical simulations in structural vibrations, rotordynamics, ecological control, etc. In recent ten years, he dedicated himself to thestudy of neurodynamics, in which the complex dynamic behaviors and mechanismsof biological neuronal firing activities and network features were investigated by thebifurcation theory, multi-scale dynamics analysis and complex networks as well asnumerical simulations. These results have been published in over three hundred re-search papers in national and international journals and six monographs and books.He has also served on several editorial boards of national and international journals.

Qishao Lu has won several prestigious prizes during his scientific career, includingthe first place of China Aerospace Science and Technology Award in 1995, the firstplace of Tianjin Natural Science Award in 2002, the second place of China NationalNatural Science Award in 2003 and others. He was elected BUAA DistinguishedProfessor in 2007 and is Honorary Professors of several universities in China, Canadaand UK.

Qishao Lu has performed dozens of grants as the primary investigator at thenational level, including several key projects from the Natural Science Foundation ofChina and the Ministry of Science and Technology of China. He has supervised andcollaborated with about fifty Ph.D students and postdocs in applied mathematicsand mechanics at BUAA. Qishao has deeply influenced many of his students andcolleagues in their academic careers. To celebrate his 70th birthday, a symposiumon the advances of methods and applications in nonlinear dynamics was held at theSouth China University of Technology, Guangzhou, China, August 7–9, 2010.

The fifteen papers herein are contributed by some of his admirers, friends andcollaborators, who come from many persuasions within the mathematical enterprise.They address and present some of advances with emphasis on newly developedtechniques that bear on further progress in nonlinear systems, as well as applicationsin biology and engineering. These fifteen papers make notable contributions to quitevarious branches of this field, which include

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• for Lagrangian equations and planar Hamiltonian systems, the fundamentalideas of the method of third order approximation are sketched and twist cri-teria are presented. Examples of stable periodic solutions are illustrated, andsome open problems are proposed (J.F. Chu, J.Z. Lei and M.R. Zhang);

• various firing patterns in the Chay neuronal model are classified by the fast-slow dynamical method and the bifurcation analysis. The relationship be-tween the bursting modes and the two-parameter bifurcation structures ofthe fast subsystem is presented (L.X. Duan, Z.Q. Yang, S.Q. Liu and D.W.Gong);

• global Hopf bifurcation analysis is explored on a six-dimensional FitzHugh-Nagumo neural network with time delay. The existence of local Hopf bifurca-tions of the system and the explicit formulae which can determine the directionof the bifurcations and the stability of the periodic solutions are obtained byusing the normal form method and the center manifold theory (F. Han, B.Zhen, Y. Du, Y.H. Zheng and M. Wiercigroch);

• a 3-D autonomous quadratic system is converted to an extended Lorenz-typesystem which contains a large class of chaotic dynamical systems. Canonicalforms are obtained with the aid of various nonsingular linear transformationsand normalization techniques (C.C. Hua, G. Chen, Q.H. Li and J.H. Ge);

• the consensus of discrete-time multi-agent systems with a directed communi-cation topology is cast into the stability of a set of matrices with the same lowdimension as that of a single agent. For neurally stable agents, it is shownthat there exists an observer-type protocol having a bounded consensus regionin the form of an open unit disk (Z.K. Li, Z.S. Duan and G. Chen);

• a three dimensional Ginzburg-Landau type equation is considered. Two fami-lies of new traveling wave solutions in term of explicit functions are presentedby using the homogeneous balance method, in which one consists of variable-amplitude solutions and the other constant-amplitude solutions (S.J. Lu, C.B.Gan, B.H. Wang, L.N. Qian and M.S. Li);

• the firing properties are introduced for the ink gland motor cells and the inkgland motor cell model. The main results including the underlying dynamicalmechanism of the firing behavior, the contribution of the two transient potas-sium currents and the calcium current, and firing control are presented (X.Y.Meng, Q.B. Ji and J. Rinzel);

• a class of generalized piecewise smooth maps is studied, which is linear at oneside and nonlinear with power dependence at the other side. The occurrenceconditions are found for border collision bifurcation and smooth fold and flipbifurcation. Different bifurcation scenarios are shown for individual cases(Z.Y. Qin, J.C. Yang, S. Banerjee and G.R. Jiang);

• The reliability of networks of beta-cells coupled by gap junctions or synapticexcitation is investigated. Simulations of the network of beta-cells reveal thatincreasing noise level decreases the reliability. Reliability will decrease whencoupling strength is small and increase when coupling strength is large (J.Y.Wang, J.Z. Su, H.P. Gonzalez and J. Rubin);

• a constraint-stabilized numerical method is presented for the planar rigidmultibody system with friction-affected translational joints, in which the slid-ers and the guides are treated as particles and bilateral constraints. The nor-mal forces of bilateral constraints are expressed by the Lagrange multipliers(Q. Wang, H.L. Peng and F.F. Zhuang);

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• the small-world Hodgkin-Huxley neuronal networks are investigated. It isshown that small delays can detriment synchronization in the network due toa dynamic clustering anti-phase synchronization transition, and the fractionof sodium and potassium channels has different impacts on spatiotemporaldynamics of neuronal networks (Q.Y. Wang, X. Shi and G. Chen);

• a p-Laplacian equation with nonlinear boundary condition is considered. Re-sults on the existence of positive solutions are obtained by the sub-supersolutionmethod and Mountain Pass Lemma (Z.D. Yang, J. Mo and S.B. Li);

• the dynamic behavior of a system of two coupled Hindmarsh-Rose neurons isconsidered. The emerging of periodic patterns is related to topological changesof its underlying bifurcations. A pathway is found from chaotic burstingbehavior to regular bursting (F. Zhang, W. Zhang, P. Meng and J. Su);

• a proportionally-fair controller with time delay is presented. The time delayis chosen to be a controllable parameter. The method of multiple scales isemployed to obtain the periodic solution arising from the Hopf bifurcation inthe congestion control model (S. Zhang and J. Xu);

• a class of nonlinear degenerate elliptic equations under the natural growth isstudied. It is proved that each bounded weak solution of A-harmonic typeequations belongs to local Holder continuity (S.Z. Zheng, X.L. Zheng and Z.Feng).

As the reader may notice, some of these contributions not only present a numberof results, ideas and techniques in physics, applied mathematics and neurodynamics,but also formulate a few open questions which may stimulate further study in theseareas. We hope that these works will help the reader to gain an adequate informationand impression of the diversity of nonlinear systems and their applications in biologyand engineering.

Finally, we would like to express our gratitude to all authors and referees fortheir critical contributions and valuable assistance. We also wish to thank AIMS,especially the DCDS-B editorial board, for providing us this opportunity to publishthis issue to honor Qishao.

Guest Editors:Zhaosheng Feng (University of Texas–Pan American, USA)

Jinzhi Lei (Tsinghua University, China)